Trying to understand differentiation/integration

gimme5minutes

As the title says Im trying to get my head around this stuff at the moment. I'm fine with the basics but after that I start having trouble. At the moment Im not sure about implicit diffentiation and integrating exponential functions. Here are two examples of problems where I just cant see how to do them.

Implicit differentiation problem -
3(x^2 + xy +y) = xy + 7y
Find the gradient to the curve using implicit differentiation.


Integration of exponential function -
S (3x -1)e^5x - (S is supposed to be the integral sign)

Anybody able to help me with this? Cheers.

LeixlipRed

Implicit Differentiation is easy enough. We use it when we cannot explicitly write y=f(x). What you do is just differentiate the expressions with just x's normally. When you differentiate y you use the Chain Rule or simpler than that, you treat y as if it were x and differentiate it then multiply by dy/dx.

Ex:

x^2 + y^2 = 3
Differentiate:
2x + 2ydy/dx = 0

Then isolate dy/dx on the left.

When x and y occur as a product you treat them as two seperate functions of x and use the Product Rule.

The next question looks like integration by parts assuming you've typed it right. That's not really easy to explain through here so just look at the wiki page for integration by parts and see if you can figure it out.

gimme5minutes

Ok cheers, so here's what Ive done for the equation 3(x^2 + xy +y) = xy + 7y

1). 3x^2 + 3xy + 3y = xy + y

2). Gather terms
3x^2 +2xy = -2y


2). Differentiate both sides, use product rule
D(3x^2) + D(2xy) = D(-2y)

=> 6x + 2xy' + 2y= -2

3). Solve for y'

2xy' = -2 - 2y- 6x

y' = (-2 - 2y - 6x)/2x

4). Substitute in (1,-1)
y' = (-2 +2 - 6)/2
y' = -3

Slope = -3


Have I got this right, is this correct?

gimme5minutes

Ive also being trying the second question regarding integrating the exponential number.

S (3x -1)e^5x dx

Here's what I've done.

1).
Let u = 5x
=> du/dx = 5
=> du=5dx
=> dx=du/5

2). Substitue in

S(3x-1)e^u . du/5

(3x-1)/5 S e^u du

3). Integrate

(3x-1)/5 e^u + K

4). Substitute back in

Final answer => (3x-1)/5 e^5x + K

Is this correct?

LeixlipRed

Second question is totally incorrect. Look up integration by parts!! First one, method looks correct. As for whether the answer is right or not I'm not sure. You seem to have turned a 7y into a y at some stage. Also, you'll have huge problems integrating if you cannot differentiate (which the other thread seems to suggest you cannot).

gimme5minutes

Yeah forgot the 7 there for some reason. I have looked up integration by parts, and that was what I got after I tried it. Could someone show me how its done, cheers.

MathsManiac

Also, when you were doing the right hand side you forgot that you were supposed to be differentiating wrt x, not y. So D(7y) gives you 7y', not just 7.

LeixlipRed

gimme5minutes wrote: »

Yeah forgot the 7 there for some reason. I have looked up integration by parts, and that was what I got after I tried it. Could someone show me how its done, cheers.

No, you're attempting integration by substitution not integration by parts. Are you sure you read it? No one will do the question for you so you need to get started yourself. Integration by parts is an adaptation of the product rule. In your question you have the product of two functions which you want to integrate. The forumla you can google. And basically you select one function and differentiate that and then integrate the other function. Trial and error on which is which should be attempted rather than me telling you which one you diff and which one you integrate.